The online servers for Halo: Reach, Halo 4, Halo 3, Halo 3: ODST, Halo: Combat Evolved Anniversary, Spartan Assault and Halo Wars will be turned off next year and digital sales of those games will end this month. The games will still work for local play, but all online services will be turned off no sooner than December of 2021.
343 has provided a chart of features for these games after the shutdown happens with matchmaking and file share ending for all the FPS games, but all local content (including custom games) will remain working. Achievements are listed as "varies" for every game and details are limited on what that means, but it does not seem promising as they later encourage players to grab all their missing achievements before the shutdown.
This shutdown will also include playing the games via backwards compatibility modes but not via Xbox One versions or MCC as those versions are running on a more modern system. However, even trying to hop onto an Xbox 360 version on Halo via your Xbox One will result in the same shutdown of online services.
Halo DLC will be available for free until the shutdown but the games themselves will no longer be listed at this point via the Xbox Store. There is supposed to be a notation in the announcement on the free DLC, but it does not appear to be listed anywhere, so your experience may vary. The 343 team has encouraged players to push for those last few achievements and to enjoy their custom file share content as much as they can, as even the legacy file share will be shut down after this move.
It has been a great run full of solid memories for the series on 360 but all good things must come to an end. Thankfully, local content will still be working, and all the games are included in the Master Chief Collection (besides Spartan Assault and Halo Wars) or otherwise available on newer platforms or PC. Oddly, Halo 2 is missing from this entire announcement although it does not appear to be listed in the store anymore either.